Capacity Spectrum Method, CSM, (ATC 40, 1996)

The Capacity Spectrum Method, CSM, was first presented by Freeman et al. (1975) as a rapid seismic assessment tool for buildings. Subsequently, the method was accepted as a seismic design tool. The steps in the method are as follows:

2.4.1.1 Nonlinear static pushover analysis of the MDOF model

A vertical distribution of the lateral loading to be applied to the structure is assumed based on the fundamental mode of vibration. Other lateral load patterns can also be used instead, section 2.3.2. A nonlinear static analysis is then carried out to give a Base Shear – Roof Displacement Curve, the Capacity Curve.

2.4.1.2 Definition of Inelastic Equivalent SDOF system, ESDOF

The capacity curve is then approximated as a bilinear relationship with the choice of a global yield point (Vy, uy) of the structural system and a final displacement (Vpi, upi). The yield point (Vy, uy) is defined such that the area A1 in Figure 2.5 is approximately equal to the area A2 in order to ensure that there is equal energy associated with each curve.

Figure 2-5 Bilinear approximation of the capacity curve

Utilising equations 2.8 and 2.9, the properties of the inelastic equivalent SDOF system, ESDOF, can be defined.

2.4.1.3 Conversion of Capacity Curve to Capacity Spectrum

The Capacity Curve is then converted to a Capacity Spectrum relationship using the following equations:

M S V

m b

a = ⋅

α ^(2.21)

ij

d PF

S u

1φ

= (2.22)

Bilinear representation Capacity spectrum

V_y

V_pi

uy upi

Displacement

Base Shear

A₁ A2

Note:

Area A1 = Area A2

Vb

u

where M is the total mass of the building, φ_ij is the modal amplitude at storey level ‘i’ for mode j, PF₁ is a participation factor and α_m is the modal mass coefficient which are given by:

2.4.1.4 Elastic Response Spectrum and Acceleration-Displacement Spectrum, ADRS format

The conversion of the capacity curve to the capacity spectrum necessitates that the elastic response or design spectrum is plotted in acceleration-displacement format, ADRS, rather than acceleration-period format, Figure 2.6. The ADRS spectrum is also denoted as the demand spectrum. This has been the first improvement of the CSM method, by Mahaney et al. (1993).

Figure 2-6 Conversion of elastic spectrum to ADRS spectrum

S_a Sa

S_d

2.4.1.5 Superposition of the Capacity Spectrum on the Elastic Damped Demand Spectrum

Once the capacity spectrum and the 5% damped elastic demand spectrum are plotted together in the ADRS format, Figure 2.7, an initial estimate of the performance point (api, dpi) using the equal displacement rule can be obtained by extending the linear part of the capacity spectrum until it intersects the 5% damped demand spectrum. Alternatively, the performance point can be assumed to be the end point of the capacity spectrum, or it might be another point chosen on the basis of engineering judgment, as ATC-40 has suggested.

Figure 2-7 Initial estimation of performance point using the Equal Displacement rule

2.4.1.6 Equivalent Viscous Damping

When structures enter the nonlinear stage during a seismic event they are subjected to damping which is assumed to be a combination of viscous damping and hysteretic damping. Viscous damping is generally accepted that is an inherent property of structures.

Hysteretic damping is the damping associated with the area inside the force-deformation Spectral Displacement Sd

Intersection point of elastic design spectrum and capacity spectrum based on equal displacement rule

Bilinear representation Capacity spectrum

ay a_pi

d_y

Elastic Design spectrum

dpi

Spectral Acceleration Sa

equivalent viscous damping is therefore associated with a specific maximum displacement dpi and is estimated using the following equation:

05 . +0

= _o

eq β

β (2.25)

Chopra (1995) has defined β0 by equating the energy dissipated in a vibration cycle of the inelastic system and of its equivalent linear system, Figure 2.8. This is provided by the following equation:

So D

E E β π

4 1

0 = (2.26)

where E_Dis the energy dissipated by damping, and E_Sois the maximum elastic strain energy.

Figure 2-8 Estimation of equivalent viscous damping using CSM method (ATC-40, 1996)

Once the maximum displacement, d_pi, has been assumed equation 2.26 becomes:

pi pi

y pi pi y

d d d

a πα

β₀ = ^{200( −}α ⁾ (2.27)

The reader is referred to ATC-40, for the derivation of equation 2.27. Other relationships have also been proposed based on ductility, µ, and strain hardening ratio, α, Chopta et al.

2000.

Using equation 2.25 the amount of damping for which the demand spectrum needs to be computed can be calculated.

2.4.1.7 Performance point of equivalent SDOF system

The new demand spectrum should then be checked if it intersects the capacity spectrum at or close enough to the estimate of performance point, Figure 2.9. If the demand spectrum intersects the capacity spectrum within an acceptable tolerance then the estimate is accepted. Otherwise the performance point is re-estimated and the procedure repeated from the step of superimposing the capacity spectrum on the ADRS spectrum, section 2.4.1.4.

Figure 2-9 Estimation of target displacement using CSM method

Intersection point of demand spectrum and capacity spectrum

Bilinear representation Capacity spectrum

ay a_pi

d_y

Demand spectrum

d_i d_pi

Note:

api ,dpi=trial performance point di=displacement intersection point

The trial performance point api ,dpi is acceptable if 0.95dpi ≤ di ≤1.05dpi

Spectral Displacement Sd

Spectral Acceleration Sa

2.4.1.8 Performance Point of MDOF system

When the performance point has been calculated it is converted to the target displacement of the MDOF system using:

u_t =PF₁φ_ijS_d (2.28)

where PF₁ is the participation factor defined in eq. 2.23 and Sd is the spectral displacement of the equivalent SDOF system defined in eq. 2.22.

The strength of structural elements and storey drifts can now be checked for the target displacement.

The notion of the CSM method can also be summarized as follows (Paret et al., 1996):

If the capacity curve can extend through the envelope of the demand curve, the structure survives the earthquake. The intersection of the capacity and demand curve represents the force and displacement of the structure for that earthquake.

An improvement of the CSM method in order to identify higher –mode effects was proposed by Paret et al. (1996), called Modal Pushover Procedure, MPP. It required several nonlinear static analyses to be carried out based on the number of modes of vibration of interest. In this way the influence of each individual mode could be observed when the individual capacity spectra were superimposed on the damped demand spectrum. However, the effects of higher modes cannot be quantified in this way since the method does not provide estimation of response, Antoniou (2002).